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* We’ve all heard of them; Why do they exist; how are they computed A work term – the amount of energy needed relative to a reference, or starting point 1000 kg Standard Energy Cost Real Energy Cost * The Physical Chemist’s secret: the activity coefficient A numerical correction applied to a model to predict properties when systems do not behave ideally The ultimate fudge factor An Activity Coefficients is a mathematical correction that describes the effects of the immediate environment on a dissolved ion. The numerical value is the quantitative deviation from ideality - a value of 1 is ideal What are the causes Ideally -Ions behave independently, in dilute solutions because they are relatively far apart Nonideally - At higher concentrations, an ion’s electric field affects other ions and changes their behavior * State 2 State 1 Work + Pure water Pure water 𝑝𝑢𝑟𝑒 𝑤𝑎𝑡𝑒𝑟 𝐺𝑆𝑡𝑎𝑡𝑒2 * 𝑝𝑢𝑟𝑒 𝑤𝑎𝑡𝑒𝑟 − 𝐺𝑆𝑡𝑎𝑡𝑒1 𝑝𝑢𝑟𝑒 𝑤𝑎𝑡𝑒𝑟 = ∆𝐺𝑤𝑜𝑟𝑘 State 2 State 1 - - + - + - - + - + + + - + + - + - - + - + Salt water + + - - - - - + + + + - - + - Salt water 𝑠𝑎𝑙𝑡 𝑤𝑎𝑡𝑒𝑟 𝑠𝑎𝑙𝑡 𝑤𝑎𝑡𝑒𝑟 𝑠𝑎𝑙𝑡 𝑤𝑎𝑡𝑒𝑟 𝐺𝑆𝑡𝑎𝑡𝑒2 − 𝐺𝑆𝑡𝑎𝑡𝑒1 = ∆𝐺𝑤𝑜𝑟𝑘 * - + - + + - + - - + + + + - - + + + - + + - + + + - - + - + + + - - + - + + Work - + - + - + - + - - + + - + + + - - - - - + - - + - - + + + - - + - + + + - + - + - + + - State 2 State 1 - - + - + - - + + + + - + + - + - + + - + - Work + - + + - - + + + - - + - + + + + - - + + + + - - + + + - + + - + - 𝐺𝑆𝑡𝑎𝑡𝑒2 − 𝐺𝑆𝑡𝑎𝑡𝑒1 = ∆𝐺 𝐸𝑥𝑐𝑒𝑠𝑠 * - - + + + - - + - + + + - - + - + - + - - + + - + + - + + + - - - - - + - - + - - + + + - - + - + + + - - + - + - + + - Ions are distributed in water. Simplify the system with a continuous dielectric medium with a swarm of charge no discrete charges Electrostatic potential r - - + - + - - + + - - + + + * + + - - + + - - - + + - + - r - + + + + - - + + + - - + Reference ion - + + + - - + - + + + - - + Volume element dV + + - Distance r from central ion + Ion cloud with charge density r * Images that explain physically why activity coefficients are needed Resistance of electrons to move at a fixed potential Material Resistance of electric field to transmit through substance Resistivity, R @20C ρ (Ω·m) Copper 10-8 Carbon steel (1010) 10-7 Sea water Drinking water Deionized water Glass Hard Rubber 𝑅(𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒) = 10 -1 10 3 10 5 10 12 10 13 Material More Insulating of current 𝑉(𝑣𝑜𝑙𝑡𝑎𝑔𝑒) 𝐼 (𝑐𝑢𝑟𝑟𝑒𝑛𝑡) * Dielectric constant, Dc @20-25C Air 1 Hard Rubber 3 Asbestos 4.8 Olive oil 3.1 Acetone 21 Ethanol 24 Ethylene glycol 37 Deionized water 80 More shielding of electric field 𝑞∗𝑄 𝑟2 𝑡𝑒𝑠𝑡 𝑐ℎ𝑎𝑟𝑔𝑒 ∗ 𝑓𝑖𝑒𝑙𝑑 𝑐ℎ𝑎𝑟𝑔𝑒 = 𝑝𝑒𝑟𝑚𝑖𝑡𝑡𝑖𝑣𝑖𝑡𝑦 ∗ 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑞 𝑎𝑛𝑑 𝑄 𝐹 𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐𝑎𝑙 𝑓𝑜𝑟𝑐𝑒, 𝑣𝑒𝑐𝑡𝑜𝑟 = 𝑘𝑒 Polarized H2O reacting to an Electric field Attenuated flux 𝐸′ 𝑥,𝑦,𝑧 Electric field flux Electron density shifts towards field 𝑬 𝑥,𝑦,𝑧 + + + + + electrons shift towards the positive charge, creating an internal electric field EH2O A dielectric material is one that polarizes in the presence of an electric field * A metal ion is a small “point charge” and produces an electric field H2O molecules orient “coordinate” around it 0.7 Å 3.5 Å. Slice it in half… M+z M+z M+z is relatively small in size, only six water can fit around it (in a sphere of course). * Layer Na+ * 1 2 3 4 5 6 7 8 9 Inner layer radius Å 0.8 3.5 6.2 8.9 11.6 14.3 17.0 19.7 22.4 Outer layer Layer radius volume Å nm3 3.5 0.2 6.2 0.8 8.9 1.9 11.6 3.6 14.3 5.7 17.0 8.3 19.7 11.4 22.4 15.0 25.1 19.1 H2O in layer N 6 27 64 119 189 276 380 500 636 H2O in sphere N 6 33 97 216 405 681 1061 1561 2197 * Na+ distance Cl- * Start with 1m3 deionized water Add 0.1 mol NaCl Calculate the ion population density 𝑚𝑜𝑙𝑒 𝑁𝑎𝐶𝑙 𝑚𝑜𝑙𝑒 𝑖𝑜𝑛 𝑖𝑜𝑛 1𝑚3 23 ∗ 2 ∗ 6.023𝑥10 ∗ 𝑚3 𝑚𝑜𝑙𝑒 𝑁𝑎𝐶𝑙 𝑚𝑜𝑙𝑒 𝑖𝑜𝑛 1027 𝑛𝑚3 𝑖𝑜𝑛 = 1.20𝑥10−7 𝑛𝑚3 0.1 Invert to obtain the volume per ion 1.20𝑥10−7 𝑖𝑜𝑛 𝑛𝑚3 6 → 8.30𝑥10 𝑛𝑚3 𝑖𝑜𝑛 Calculate spherical radius of this volume 6 8.30𝑥10 𝑟= * 4∗𝜋 𝑛𝑚3 𝑖𝑜𝑛 3 1 3 = 125 𝑛𝑚 NaCl conc, NaCl conc, Inter-ion H2O Act Coef, mol/m3 mol/l distance, molecules nm per ion 0 0 1 1e-6 1e-9 579 2.78e10 1.000 1e-5 1e-8 270 2.78e9 1.000 1e-4 1e-7 125 2.78e8 1.000 1e-3 1e-6 58 2.78e7 0.999 1e-2 1e-5 27 2.78e6 0.996 1e-1 1e-4 12.5 2.78e5 0.988 1 0.001 5.8 27775 0.965 10 0.01 2.7 2778 0.902 100 1000 0.1 1 1.25 0.58 278 28 0.771 0.644 * * * Ions can be visualized as point charge surrounded by water * Water shields the electric fields created by each ion * In dilute solutions, the ions are too far apart for the electric fields to overlap Distance=250 nm Na+ r=125 nm r=125 nm There is no interaction between the two ions, they are too far apart Sr+2 =the individual ion * =the water sheath or the water oriented around the ion δ- =partial charge around the water sheath Cl- When concentration increases, ions distance decreases. Electric fields start to overlap other ions and the water surrounding them δ+ δ+ δ+ δ+ Sr+2 δ+ δ+ Sr+2 δ+ δ+ δ+ δ+ Electric fields start to overlap δ+ δ- δ- δ- This changes ion behavior δ- δ- δ+ δ- δ- δδSO4-2 δδ- SO4-2 δ- * * This electric field (electrostatic) effect is quantified using an activity coefficient : z i e 2 i A I δ+ δ+ δ+ Sr+2 Where: δ+ + δ+ δ δ+ The charges are starting to interact δ- 1 I ionic strength z i2 mi 2 mi concentration of ion δ+ Sr+2 + + δ+ δ δ zi charge on ion A constant δ+ δ- δ- δ - δ- δ δδ- - SO4-2 SO4-2 δ- This is called the Debye Huckel model * δ- δ- δ- Most common model for estimating activity coefficients Assumes salts are completely ionized (limiting law) Valid up to 0.01 mole/L salinity Equation is -logγ± = Az+z-√I where: o γ± = mean of the activity coefficients for the + and - ions; o A is a constant that depends on temperature and the dielectric constant ε (A = 1.825 x 106(εT)-3/2 = 0.51 at 25◦ C in water); o o z+ and z- are the + and - ion charges I is the ionic strength. * 21 *Activity coefficients becomes part of the equilibrium equation 𝐴+𝐵 =𝐶 𝛾𝐶 ∗ 𝐶 𝐾= 𝛾𝐴 ∗ 𝐴 ∗ 𝛾𝑏 ∗ 𝐵 *In dilute solutions, 1 , and 𝐶 𝐾= 𝐴 ∗ 𝐵 * Ionic strength is a measure of the system’s electric field. 𝐼 1 = 2 𝑚𝑖 𝑧𝑖 2 Where 𝑖 mi = ion concentration & zi = ion charge Example A 1.0 mol/kg solution of NaCl has 1.0 moles of Na+1 ion and 1.0 moles of Cl-1 ion per Kg H2O. I 1 2 2 Z Na 1 mNa 1 Z Cl 1 mCl 1 2 I The ionic strength is 1.0 molal. * 1 2 1 1.0 12 1.0 1.0 2 A 1.0 mol/kg solution of CaSO4 has 1.0 moles of Ca+2 ion and 1.0 moles of SO4-2 ion per Kg H2O. 1 2 2 I Z Ca 2 mCa 2 Z SO 42 mSO 42 2 1 2 2 I 2 1.0 2 1.0 4.0 2 The ionic strength is 4.0 mol * Debye-Huckel model (1922), most common and easiest ln i zi2 A I Good to about 0.01 mol/L concentration Extended Debye-Huckel model – extends the concentration limits I ln i z A bI 1 I Good to about 0.1 mol/L 2 i Davies Equation (1938) – a further extension Good to about 0.3 mol/L * 27 Pitzer Equation 2 ln | Z Z | f m 1.5 2 2 B m C Bromley-Zematis A | Z Z | Equation I 0.06 0.6 B | Z Log 1 I Helgeson A | Z Z Log i l | I 1 a0 B I Z | I 1.5 1 | Z Z | I k k 2 i ,k b Y I k k k k BI CI 2 DI 3 bil Yl I l ,k l l k * All are good to between 10 and 30 mol/kg H2O. The problem is trying to use them… 28 bil Yi I i i To get better activity coefficients, we need the more complex models Complex models cannot be solved by hand (they are non-linear and need multiple iterations) Thus freeware and commercial software products were developed * 29